The invention relates, in general terms, to recent cryptographic techniques and the processes used therein.
More specifically, the invention relates, according to a first -aspect, to a cryptographic process using an elliptic curve represented in a plane by means of an equation containing first and second parameters (a, b), a bilinear matching, and calculations in a finite group of integers constructed around at least one first reduction rule reducing each integer to its remainder in a whole division by a first prime number (p) that constitutes a third parameter, the elements of the finite group being in bijection with points selected on the elliptic curve, and the number of which is linked to a fourth parameter (q), this process using public and private keys, each of which is represented by a given point of the elliptic curve or by a multiplication factor between two points of this curve.
Such processes are the basis for the most recent cryptographical techniques, based on a bilinear pairing, for example a Weil pairing.
A notable example of such a process is provided in the article entitled “Identity-Based Encryption from the Weil Pairing”, published in 2003 in the SIAM Journal of Computing, Volume 32, No. 3, pages 586 to 615 by Dan Boneh and Matthew Franklin.
Current processes of this type rely on the use of supersingular elliptic curves.
And yet, the use of such curves implies complex calculations, which may be an obstacle to the use of these processes in all cases in which the available calculation capacities are limited.